Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

Haus, Emanuele; Maspero, Alberto

doi:10.1016/j.jfa.2019.108316

Cited by 13 publications

(8 citation statements)

References 33 publications

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“…At this level of generality such results are in fact optimal as showed in [Bou99b]. See also [Mas18], [HM19] for examples of growth. A parallel point of view is to study the reducibility of Schrödinger operators with quasi-periodic potentials by requiring stronger non-resonance conditions on the frequency, see [EK09].…”

Section: Introductionmentioning

confidence: 69%

Linear Schrödinger equation with an almost periodic potential

Montalto¹,

Procesi²

2019

Preprint

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We study the reducibility of a Linear Schrödinger equation subject to a small unbounded almostperiodic perturbation which is analytic in time and space. Under appropriate assumptions on the smallness, analiticity and on the frequency of the almost-periodic perturbation, we prove that such an equation is reducible to constant coefficients via an anaytic almost-periodic change of variables. This implies control of both Sobolev and Analytic norms for the solution of the corresponding Schrödinger equation for all times. *We shall be particularly interested in almost-periodic functions where X = H(T σ )is the space of analytic functions T σ → C, where T σ := {ϕ ∈ C : Re(ϕ) ∈ T, |Im(ϕ)| ≤ σ} is the thickened torus. Now we are ready to state precisely our main result. We make the following assumptions.• (H1) The functions V 0 , V 1 , V 2 are almost-periodic and analytic, in the sense of Definition 1.3, for σ > 0 and X = H(T σ ). • (H2) We assume that (1.6)

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Section: Introductionmentioning

confidence: 69%

Linear Schrödinger equation with an almost periodic potential

Montalto¹,

Procesi²

2019

Preprint

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“…In [ANS19], the authors prove exponential growth of the energy norm for a linear (and nonlinear) harmonic oscillator perturbed by the angular momentum operator (see [ANS19, Theorem 4.5]). We refer to [HM20,Mas19] for more examples with growth of norms and to [BGMR21] for bounds on abstract linear Schrödinger equations. Let us mention the article [LZZ21] in which the authors obtain very precise results on the dynamics of a family of perturbations of the harmonic oscillator.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Growth of Sobolev norms for linear Schrödinger operators

Thomann¹

2021

Annales Henri Lebesgue

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We give an example of a linear, time-dependent, Schrödinger operator with optimal growth of Sobolev norms. The construction is explicit, and relies on a comprehensive study of the linear Lowest Landau Level equation with a time-dependent potential.Résumé. -Nous donnons un exemple d'un opérateur de Schrödinger linéaire, dépendant du temps, avec une croissance optimale des normes de Sobolev. La construction est explicite, et s'appuie sur une étude complète de l'équation linéaire de plus bas niveau de Landau avec un potentiel dépendant du temps.

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“…Finally recently Faou-Raphael [22] constructed a transporter for the harmonic oscillator on R which is a time dependent function (and not a pseudodifferential operator), and Thomann [44] has constructed a transporter for the harmonic oscillator on the Bargman-Fock space. Finally we recall the long-time growth result [25] for the semiclassical anharmonic oscillator on R d .…”

Section: Introductionmentioning

confidence: 98%

Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon

Maspero¹

2021

Preprint

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In this paper we consider linear, time dependent Schrödinger equations of the form i∂tψ = K0ψ + V (t)ψ, where K0 is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, and V (t) a time periodic potential. We give sufficient conditions on V (t) ensuring that K0 +V (t) generates unbounded orbits. The main condition is that the resonant average of V (t), namely the average with respect to the flow of K0, has a nonempty absolutely continuous spectrum and fulfills a Mourre estimate. These conditions are stable under perturbations. The proof combines pseudodifferential normal form with dispersive estimates in the form of local energy decay. We apply our abstract construction to the Harmonic oscillator on R and to the half-wave equation on T; in each case, we provide large classes of potentials which are transporters.

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Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

Cited by 13 publications

References 33 publications

Linear Schrödinger equation with an almost periodic potential

Linear Schrödinger equation with an almost periodic potential

Growth of Sobolev norms for linear Schrödinger operators

Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon

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